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Home Page Title Page Contents Page 1 of 49 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit STAT231: STATISTICS Paul Marriott [email protected] May 8, 2008
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Home Page Title Page Contents Page 2 of 49 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Chapter 3: Models and the Likelihood function In this chapter we will look at the mathematical structure of a number of statistical models. The actual process of fitting and checking that the model is appropriate will be looked at in the following Chapter 5. One important aim of this chapter is to be clear what the math- ematical parts of each model mean relative to the general PP- DAC structure of the statistical method.
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Home Page Title Page Contents Page 3 of 49 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Random variables The basic tool from probability theory which we shall use in this chapter is the idea of a random variable. a number whose value is determined by chance. We characterise the behaviour of a random variable by its dis- tribution Look at a small number of parametric families of these distri- butions discrete (for example, binomial or Poisson) continuous (for example, Gaussian or exponential).
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Home Page Title Page Contents Page 4 of 49 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Random variables It is important to clearly differentiate between a random variable X and a set of independent realisations x 1 , . . . , x n from it. A random variable as a variable whose possible values are numerical outcomes of a ‘random experiment’. The random variable X has a distribution using which we can make probability statements, such as P ( X > 1) . A particular realisation of X is the result of running the ex- periment a particular time and getting a particular results x . In general we write random variables in capital letters (e.g. X ) and realisations in lowercase ( x ). We can have many independent realisations (i.e. repeat the experiment many times) of a fixed random variable which we can denote by x 1 , x 2 , · · · , x n . Very informally, in terms of PPDAC, we are going to let the random variable represent the process of taking a sample, while the realisation will represent an actual sample.
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Home Page Title Page Contents Page 5 of 49 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Realisations in R The G(0,1) random variable is function r norm(1, mean=0, sd=1)) A realisation is the result of running code > rnorm(1, mean=0, sd=1) [1] 0.4533667 > rnorm(1, mean=0, sd=1) [1] -1.517403 Can get 10 independent realisations > rnorm(10, mean=0, sd=1) [1] 0.6081998 0.5184049 0.5394318 [3] 1.4964474 1.7157859 -0.4517295 [7] 1.3502587 1.6516654 0.4863180 [10] -0.1426848 Review Keywords: Random variables, parametric families of dis- tributions, realisation, coding in R
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